Macroscopic averages in Abinit

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Macroscopic averages in Abinit Université de Liège Macroscopic averages in Abinit Javier Junquera Départament de Physique Université de Liège

Macroscopic averages Microscopic quantities Macroscopic quantities Very rapidly functions of position Resembles the microscopic atomic structure Output of first-principles calculations (DEN, POT, …) Smooth variations with the position Magnitudes that enter in the discussion of dielectrics. Magnitudes measured experimentally. Macroscopic Average Technique Average over a region that is small on the macroscopic scale, but large compared with the atomic dimensions.

Macroscopic average technique First step: Planar average Second step: Averaging with a filter (convolution with step functions) A. Baldereschi et al., Phys. Rev. Lett., 61, 734 (1988) L. Colombo et. al., Phys. Rev. B, 44, 5572 (1991)

Band offsets BAND OFFSET: Relative position of the band gaps of both constituents at the interface CBO: Conduction Band Offset VBO: Valence Band Offset <VA,B> Arbitrary averages of the electrostatic potential Relation <VA> <VB>? <VA> <VB>

Band offset calculation Band-structure term Difference between the two valence band edges Standard bulk structure calculations Line-up of the electrostatic potential Relates the average of the electrostatic potential in the two materials. Can only be obtained from a SCF calculation on the supercell Contains all the intrinsic effect of the interface Obtained by macroscopic average technique A. Baldereschi, S. Baroni and R. Resta, PRL 61, 734 (1988)

Double-macroscopic average Planar average Average on normal direction Convolution with two filter functions L. Colombo et al, Phys. Rev. B, 44, 5572 (1991)

BaO/BaTiO3 band offset ABINIT: SIESTA: 3 BaO / 3 BaTiO3 VBO: -0.31 eV

Schottky barrier of SrRuO3/BaTiO3 SrO-TiO2 interface ABINIT: 3.5 SrRuO3 / 2.5 BaTiO3 fp: -1.75 eV

Dynamical charges Collective displacement: macroscopic polarization P Linear response : Individual displacement along z: local dipole p - from the density : Finite difference : - from the potential : R. Martin and K. Kunc, PRB 24, 2081 (1981) Layer dependent quantity relevant for thin films: Z*L

Computation of longitudinal charges Dt -Dt Dr(z) DV(z) DV

BaO/BaTiO3 interface: Z*L along z Z*L(Ba)= 0.65 Bulk BO Z* L(O||)= -0.65 Z* L = 1.06 Z* L = 0.44 Z* L = 0.99 Z* L = 0.50 Z* L = 0.68 Z* L = 0.64 Z*L = 0.64 Ba Ti O O2 Z* L = -0.33 Z* L = -0.83 Z* L = -0.32 Z* L = -0.72 Z* L = -0.62 Z* L = -0.64 linear response and finite differences techniques yield similar results sharp interface (for Z*) Z* L(Ba) = 0.43 Bulk BTO Z* L(O||) = -0.83 Z* L(Ti) = 1.06 Z* L(O^) = -0.33

SrRuO3/BaTiO3 interface: Z*L along z Z*L(Sr,Ru)= 0 Bulk SRO Z* L(O||)= 0 Sr Ti Ru Ba O2 O Z*L = 0.00 Z* L = -0.00 Z* L = ??? Z* L = -0.04 only finite differences technique broad interface (for Z*) Z* L = -0.18 Z* L = ??? Z* L = -0.48 Z* L = 0.26 Z* L = -0.52 Z* L = 0.97 Z* L = -0.47 Z* L = 0.53 Z* L = -0.33 Z* L = 1.06 Z* L(Ba) = 0.43 Bulk BTO Z* L(O||) = -0.83 Z* L(Ti) = 1.06 Z* L(O^) = -0.33

Electronic dielectric constants F. Bernardini & V. Fiorentini, Phys. Rev. B, 58 15292 (98) Converge: Size of the supercell Atomic displacemets